Fluid types (e.g., oil, water, gas) may exist in a variety of materials, including geological formations, and can be identified using nuclear magnetic resonance (NMR) diffusion modulated amplitude (DMA) data to compute a probability density function, which may be expressed as a graph illustrating porosity as a function of diffusion in cm2/sec and transverse relaxation time in milliseconds (e.g., using a what is known to those of skill in the art as a “T2-D map”). Interested readers may refer to U.S. Pat. No. 6,512,371 (incorporated herein by reference in its entirety) describing how nuclear magnetic resonance DMA data are acquired.
Due to the huge amounts of data typically involved in calculating the function, reduction algorithms are commonly applied to reduce the computational intensity of the problem, which may take the general form: M(tD,t)=∫∫κ2(t,T2)·ƒ(D,T2)·κ1(tD,D)dDdT2. Interested readers may refer to U.S. Pat. No. 6,462,542 (incorporated herein by reference in its entirety) describing how T2-D maps may be calculated using a compression algorithm.
For example, incorporating the singular value decomposition of kernel matrices K1 (the discrete diffusion kernel) and K2 (the discrete relaxation kernel) into a compression algorithm, along with selecting a threshold condition number for the matrix K0=K1K2, yields a projection of the data onto the range space while preserving the Frobenius norm of the data matrix. Unfortunately, once the original data is compressed, the original values are lost. In addition, this method is limited to using a zero-order regularization matrix for the probability density function, and independent regularization of the transverse relaxation time constant distribution and diffusion distribution is not possible.